Lp加Lpq正则极小化问题临近型算法研究

【作者】 王丽华；

【导师】 彭定涛；

【作者基本信息】 贵州大学 ， 数学， 2025， 硕士

【摘要】 近年来,随着大数据技术的快速发展和数据结构的复杂性不断提升,传统稀疏模型在应对高维数据分析时逐渐显现出两方面缺陷:其一,基于稀疏性建模未能有效捕捉变量间的内在结构特征;其二,仅基于稀疏先验信息或组稀疏先验信息难以兼容实际应用中同时存在的稀疏性与组稀疏性特征.为克服上述缺陷,混合稀疏优化问题逐渐成为研究热点,且在许多领域展现出显著性优势.混合稀疏优化问题通过整合控制稀疏性特征的正则项与组稀疏性特征的正则项,在保持变量选择能力的同时又显式建模结构化稀疏特征.因此,本文针对一类混合稀疏优化问题展开研究,并提出基于临近算子解析解的交替极小化算法与交替方向乘子法.本文研究l_q加l_p,q正则优化模型,其目标函数由梯度Lipschitz连续的损失函数、稀疏惩罚项（l_q范数）与组稀疏惩罚项(l_p,q范数)共同构成,其中0<q<1,p∈{1,2}.针对目标函数非凸、非光滑及非Lipschitz连续的特性,通过引入辅助变量建立了等式约束优化问题.考虑到交替方向乘子法在非凸非光滑设置下收敛性保证的不足,本文从两个理论角度推进研究:（1）基于二次罚函数框架,提出线性临近交替极小化算法（Linearized Proximal Alternating Minimization Algorithm,LPA）,证明 了算法生成迭代序列在Kurdyka-Lojasiewicz（KL）性质下全局收敛到罚函数问题的Critical点,并建立其收敛速率估计;（2）根据线性临近交替方向乘子法（Linearized Proximal Alternating Direction Method of Multipliers,LPADMM）,在温和条件下,证明算法生成的迭代点列的极限点为原问题的稳定点.数值实验部分分别通过数值模拟实验与真实图像重建实验来系统评估LPA,LPADMM算法的性能.实验结果表明:LPA与LPADMM算法均能更为有效求解混合稀疏优化问题且更具鲁棒性.本文还考虑部分稀疏部分组稀疏优化问题,其中损失函数为梯度Lipschitz连续函数,惩罚项为基于稀疏和组稀疏先验性下的范数组合,稀疏部分为关于变量x的l_q范数,组稀疏部分为关于变量y的l_p,q范数,即λ₁‖x‖_q^q+λ₂ ‖y‖_qq^p,其中0<q<1,p ∈ {1,2}.结合目标函数的可分结构特性提出求解部分稀疏部分组稀疏优化问题的临近交替极小化算法（Proximal Alternating Minimization Method,PAM）.在温和条件下,证明了算法生成子列收敛到部分稀疏部分组稀疏优化问题的Critical点,并基于KL理论框架得到算法的全局收敛性及收敛速率.数值实验部分对PAM算法的实际性能进行评估.实验结果表明:PAM算法在处理部分稀疏部分组稀疏优化问题时更具有效性与鲁棒性.更多还原

【Abstract】 In recent years,with the rapid development of big data technology and the increasing complexity of data structure,the traditional sparse model gradually shows two defects when dealing with high-dimensional data analysis.First,sparsity-based modelling fails to effectively capture the intrinsic structural features among variables.Secondly,based on sparse prior information or group sparse prior information merely,it is difficult to be compatible with both sparsity and group sparsity features in practical applications.In order to overcome the above shortcomings,mixed sparse optimization problems have gradually become a popular research topic,and have shown significant advantages in many fields.Mixed sparse optimization problems explicitly model structured sparse features while maintaining the variable selection ability by integrating the regular terms of control sparsity features and group sparsity features.Therefore,in this paper,we study a class of mixed sparse optimization problems and propose a alternating minimization algorithm and a alternating direction multiplier algorithm based on the analytic solution of proximal operators.In this paper,we study l_q plus l_p,q regular optimization model,whose objective function consists of a gradient Lipschitz continuous loss function,a sparse penalty term（l_q norm）together with a group sparse penalty term(l_p,q norm),where 0<q<1,p ∈ {1,2}.For the non-convex,non-smooth and non-Lipschitz continuous characteristics of the objective function,the equation constrained optimization problem is established by introducing auxiliary variable.Considering the lack of convergence guarantee of the alternating direction multiplier method in non-convex and non-smooth settings,this paper promotes the research from two theoretical perspectives:（1）Based on the quadratic penalty function framework,we propose the linearized proximal alternating minimization algorithm（LPA）,prove the global convergence of the algorithm generating iterative sequences to the Critical point of the penalty function under the Kurdyka-Lojasiewicz（KL）property,and establish an estimate of the rate of convergence of the algorithm.（2）According to the method of linearized proximal alternating direction multipliers（LPADMM）.Under mild conditions,it is proved that the limit points of the iterative point sequence generated by the algorithm are the stationary points of the original problem.Numerical experiments are conducted to systematically evaluate the performance of LPA and LPADMM through simulation and real image reconstruction experiment.Experimental results show that both LPA and LPADMM algorithms are more efficient and robust in solving mixed sparse optimization problems.In this paper,we also consider the partial sparse partial group sparse optimization problem,where the loss function is a gradient Lipschitz continuous function,the penalty term is based on the combination of norms under the sparse and group sparse priors,the sparse part is the l_q norm with respect to the variable x,and the group sparse part is the l_p,q norm with respect to the variable y,i.e.,λ₁‖x‖_q^q+λ₂ ‖y‖_qq^p norms.where 0<q<1,p ∈ {1,2}.By exploiting the separability structure property of the objective function,the proximal alternating minimization algorithm（PAM）is proposed for solving partial sparse partial group sparse optimization problems.Under mild conditions,it is proved that the generating subsequence of the algorithm converges to the Critical point of the partial sparse partial group sparse optimization problem,and the global convergence of the algorithm and the rate of convergence are obtained based on the theoretical framework of KL.The numerical experiments also assess the performance of the PAM algorithm.Experimental results show that the PAM algorithm is more effective and robust in handling partial sparse partial group sparse optimization problems.更多还原